STAR FORMATION IN ISOLATED DISK GALAXIES. II. …more, inefﬁcient star formation can be found well outside the threshold radius (Ferguson et al. 1998). What is the origin of the

STAR FORMATION IN ISOLATED DISK GALAXIES. II. SCHMIDT LAWSAND EFFICIENCY OF GRAVITATIONAL COLLAPSE

Yuexing Li,1,2, 3

Mordecai-Mark Mac Low,1,2

and Ralf S. Klessen4

Received 2005 August 1; accepted 2005 November 4

ABSTRACT

We model gravitational instability in a wide range of isolated disk galaxies, using GADGET, a three-dimensional,smoothed particle hydrodynamics code. The model galaxies include a dark matter halo and a disk of stars andisothermal gas. Absorbing sink particles are used to directly measure the mass of gravitationally collapsing gas.Below the density at which they are inserted, the collapsing gas is fully resolved. We make the assumption that starsand molecular gas form within the sink particle once it is created and that the star formation rate is the gravitationalcollapse rate times a constant efficiency factor. In our models, the derived star formation rate declines exponentiallywith time, and radial profiles of atomic and molecular gas and star formation rate reproduce observed behavior. Wederive from our models and discuss both the global and local Schmidt laws for star formation: power-law relationsbetween surface densities of gas and star formation rate. The global Schmidt law observed in disk galaxies isquantitatively reproduced by our models. We find that the surface density of star formation rate directly correlateswith the strength of local gravitational instability. The local Schmidt laws of individual galaxies in our modelsshow clear evidence of star formation thresholds. The variations in both the slope and the normalization of thelocal Schmidt laws cover the observed range. The averaged values agree well with the observed average and with theglobal law. Our results suggest that the nonlinear development of gravitational instability determines the localand global Schmidt laws and the star formation thresholds. We derive from our models the quantitative dependence ofthe global star formation efficiency on the initial gravitational instability of galaxies. The more unstable a galaxy is, thequicker and more efficiently its gas collapses gravitationally and forms stars.

Subject headings: galaxies: evolution — galaxies: ISM — galaxies: kinematics and dynamics — galaxies: spiral —galaxies: star clusters — stars: formation

1. INTRODUCTION

Stars form at widely varying rates in different disk galaxies(Kennicutt 1998b). However, they appear to follow two simpleempirical laws. The first is the correlation between the star for-mation rate (SFR) density and the gas density, the ‘‘Schmidtlaw’’ as first introduced by Schmidt (1959):

�SFR ¼ A�Ngas; ð1Þ

where �SFR and �gas are the surface densities of SFR and gas,respectively.

When�gas and�SFR are averaged over the entire star-formingregion of a galaxy, they give rise to a global Schmidt law.Kennicutt (1998a) found a universal global star formation law ina large sample that includes 61 normal spiral galaxies that haveH�, H i, and CO measurements and 36 infrared-selected star-burst galaxies. The observations show that both the slope N �1:3 1:5 and the normalization A appear to be remarkably con-sistent from galaxy to galaxy. There are some variations, how-ever. For example, Wong & Blitz (2002) reported N � 1:1 1:7for a sample of seven molecule-rich spiral galaxies, dependingon the correction of the observed H� emission for extinction in

deriving the SFR. Boissier et al. (2003) examined 16 spiralgalaxies with published abundance gradients and found N �2:0. Gao & Solomon (2004b) surveyed HCN luminosity, a tracerof dense molecular gas, from 65 infrared or CO-bright galaxiesincluding nearby normal spiral galaxies, luminous infrared gal-axies, and ultraluminous infrared galaxies. Based on this survey,Gao & Solomon (2004a) suggested a shallower star formationlaw with a power-law index of 1.0 in terms of dense moleculargas content.

When �gas and �SFR are measured radially within a galaxy,a local Schmidt law can be measured. Wong & Blitz (2002) in-vestigated the local Schmidt laws of individual galaxies in theirsample. They found similar correlations in these galaxies, but thenormalizations and slopes vary from galaxy to galaxy, with N �1:2 2:1 for total gas, assuming that extinction depends on gascolumn density (or N � 0:8 1:4 if extinction is assumed con-stant). Heyer et al. (2004) reported that M33 has a much deeperslope, N ’ 3:3.

The second empirical law is the star formation threshold. Starsare observed to form efficiently only above a critical gas surfacedensity. Martin&Kennicutt (2001) studied a sample of 32 nearbyspiral galaxies with well-measured H� and H2 profiles and de-monstrated clear surface density thresholds in the star formationlaws in these galaxies. They found that the threshold gas density(measured at the outer threshold radius where SFR drops sharply)ranges from 0.7 to 40 M� pc�2 among spiral galaxies, and thethreshold density for molecular gas is�5–10M� pc�2. However,they found that the ratio of gas surface density at the thresholdto the critical density for Toomre (1964) gravitational instability,�Q ¼ �gas/�crit, is remarkably uniform with �Q ¼ 0:69 � 0:2.They assumed a constant velocity dispersion of the gas, the

1 Department of Astronomy, Columbia University, 1328 Pupin Hall, 550West 120th Street, New York, NY 10027.

2 Department of Astrophysics, American Museum of Natural History, 79thStreet at Central Park West, New York, NY 10024-5192; [email protected].

3 Current address: Harvard-Smithsonian Center for Astrophysics, 60 GardenStreet, Cambridge, MA 02138; [email protected].

4 Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482Potsdam, Germany; [email protected].

879

The Astrophysical Journal, 639:879–896, 2006 March 10

# 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.

effective sound speed, of cs ¼ 6 km s�1. Such a density thresholdapplies to normal disk galaxies (e.g., Boissier et al. 2003), ellip-tical galaxies (e.g., Vader & Vigroux 1991), low surface bright-ness galaxies (van der Hulst et al. 1993), and starburst galaxies(Elmegreen 1994). However, there are a few exceptions, such asdwarf and irregular galaxies (e.g., Hunter et al. 1998). Further-more, inefficient star formation can be found well outside thethreshold radius (Ferguson et al. 1998).

What is the origin of the Schmidt laws and the star formationthresholds? The mechanisms that control star formation in gal-axies, such as gravitational instability, supersonic turbulence,magnetic fields, and rotational shear are widely debated (Shuet al. 1987; Elmegreen 2002; Larson 2003; Mac Low & Klessen2004). At least four types of models are currently discussed. Thefirst type emphasizes self-gravity of the galactic disk (e.g., Quirk1972; Larson 1988; Kennicutt 1989, 1998a; Elmegreen 1994). Inthese models, the Schmidt laws do not depend on the local starformation process but are simply the results of global gravita-tional collapse on a free-fall time. In the second type, the globalSFR scales with either the local dynamical time, invoking cloud-cloud collisions (e.g., Wyse 1986; Wyse & Silk 1989; Silk 1997;Tan 2000), or the local orbital time of the galactic disk (e.g.,Elmegreen 1997; Hunter et al. 1998). A third type, which in-vokes hierarchical star formation triggered by turbulence, hasbeen proposed by Elmegreen (2002). In this model, the Schmidtlaw is scale-free, and the SFR depends on the probability dis-tribution function (PDF) of the gas density produced by galacticturbulence, which appears to be lognormal in simulations ofturbulent molecular clouds and interstellar medium (e.g., Scaloet al. 1998; Passot & Vazquez-Semadeni 1998; Ostriker et al.1999; Klessen 2000;Wada&Norman 2001; Ballesteros-Paredes& Mac Low 2002; Padoan & Nordlund 2002; Li et al. 2003;Kravtsov 2003; Mac Low et al. 2005). Recently, Krumholz &McKee (2005) extended this analysis with additional assumptionssuch as the virialization of themolecular clouds and star formationefficiency (SFE) to derive the SFR from the gas density PDFs.They successfully fitted the global Schmidt law, but their theorystill contains several free or poorly constrained parameters anddoes not address the observed variation in local Schmidt lawsamong galaxies. A fourth type appeals to the gas dynamics andthermal state of the gas to determine the star formation behavior.Struck-Marcell (1991) and Struck & Smith (1999), for example,suggest that galactic disks are in thermohydrodynamic equilib-rium maintained by feedback from star formation and counter-circulating radial gas flows of warm and cold gas.

There is considerable debate on the star formation threshold aswell. Martin & Kennicutt (2001) suggest that the threshold den-sity is determined by the Toomre criterion (Toomre 1964) forgravitational instability. Hunter et al. (1998) argued that the crit-ical density for star formation in dwarf galaxies depends on therotational shear of the disk. Wong & Blitz (2002) claimed noclear evidence for a link between �Q and star formation. Instead,they suggested that �Q is a measurement of gas fraction. Boissieret al. (2003) found that the gravitational instability criterion haslimited application to their sample. Note that all the modelsabove are based on an assumption of constant sound speed forthe gas. Schaye (2004) proposed a thermal instability model forthe threshold, in which the velocity dispersion or effective tem-perature of the gas is not constant, but drops from a warm (i.e.,104 K) to a cold phase (below 103 K) at the threshold. He sug-gested that such a transition is able to reproduce the observedthreshold density.

While each of these models has more or less succeeded inexplaining the Schmidt laws or the star formation threshold, a

more complete picture of star formation on a galactic scale re-mains needed. Meanwhile, observations of other properties re-lated to star formation in galaxies have provided more clues tothe dominant mechanism that controls global star formation.An analysis of the distribution of dust in a sample of 89 edge-

on, bulgeless disk galaxies by Dalcanton et al. (2004) shows thatdust lanes are a generic feature of massive disks with Vrot >120 km s�1 but are absent in more slowly rotating galaxies withlower mass. These authors identify the Vrot ¼ 120 km s�1 tran-sition with the onset of gravitational instability in these galaxiesand suggest a link between the disk instability and the formationof the dust lanes that trace star formation.Color gradients in galaxies help trace their star formation his-

tory by revealing the distribution of their stellar populations(Searle et al. 1973). A comprehensive study of color gradients in121 nearby disk galaxies by Bell & de Jong (2000) shows thatthe star formation history of a galaxy is strongly correlated withthe surface mass density. Similar conclusions were drawn byKauffmann et al. (2003) from a sample of over 105 galaxies fromthe Sloan Digital Sky Survey. Recently, MacArthur et al. (2004)carried out a survey of 172 low-inclination galaxies spanningHubble types S0–Irr to investigate optical and near-IR color gra-dients. These authors find strong correlations in age and metal-licity with Hubble type, rotational velocity, total magnitude, andcentral surface brightness. Their results show that early-type,fast rotating, luminous, or high surface brightness galaxies ap-pear to be older and more metal-rich than their late-type, slowrotating, or low surface brightness counterparts, suggesting anearly and more rapid star formation history for the early-typegalaxies.These observations show that star formation in disks corre-

lates well with the properties of the galaxies such as rotationalvelocity, velocity dispersion, and gas mass, all of which directlydetermine the gravitational instability of the galactic disk. Thissuggests that, on a galactic scale, gravitational instability con-trols star formation.The nonlinear development of gravitational instability and its

effect on star formation on a galactic scale can be better under-stood through numerical modeling. There have been many sim-ulations of disk galaxies, including isolated galaxies with variousassumptions of the gas physics and feedback effects (e.g., Thacker& Couchman 2000; Wada & Norman 2001; Noguchi 2001;Barnes 2002; Robertson et al. 2004; Li et al. 2005b, hereafterPaper I; Okamoto et al. 2005), galaxy mergers (e.g., Mihos &Hernquist 1994; Barnes & Hernquist 1996; Li et al. 2004), andgalaxies in a cosmological context, with different assumptionsabout the nature and distribution of dark matter (e.g., Katz &Gunn 1991; Navarro & Benz 1991; Katz 1992; Steinmetz &Mueller 1994; Navarro et al. 1995; Sommer-Larsen et al. 1999,2003; Steinmetz&Navarro 1999; Springel 2000; Sommer-Larsen& Dolgov 2001; Springel & Hernquist 2003; Governato et al.2004). However, in most of these simulations gravitational col-lapse and star formation either are not numerically resolved or arefollowed with empirical recipes tuned to reproduce the observa-tions a priori. There are only a handful of numerical studies that fo-cus on the star formation laws. Early three-dimensional smoothedparticle hydrodynamics (SPH) simulations of isolated barred gal-axies were carried out by Friedli & Benz (1993, 1995) and Friedliet al. (1994). In Friedli &Benz (1993), the secular evolution of theisolated galaxies was followed by modeling of a two-component(gas and stars) fluid, restricting the interaction between the twoto purely gravitational coupling. The simulations were improvedlater in Friedli & Benz (1995) by including star formation andradiative cooling. These authors found that their method to

LI, MAC LOW, & KLESSEN880 Vol. 639

simulate star formation, based on Toomre’s criterion, naturallyreproduces both the density threshold of 7M� pc�2 for star forma-tion and the global Schmidt law in disk galaxies. They also foundthat the nuclear starburst is associated with bar formation in thegalactic center. Gerritsen & Icke (1997) included stellar feedbackin similar two-component (gas and stars) simulations that yieldeda Schmidt law with power-law index of �1.3. However, thesesimulations included only stars and gas, and no dark matter. Morerecently, Kravtsov (2003) reproduced the global Schmidt lawusing self-consistent cosmological simulations of high-redshiftgalaxy formation. He argued that the global Schmidt law is aman-ifestation of the overall density distribution of the interstellar me-dium and that the global SFR is determined by the supersonicturbulence driven by gravitational instabilities on large scales,with little contribution from stellar feedback.However, the strengthof gravitational instability was not directly measured in this im-portant work, so a direct connection could not be made betweeninstability and the Schmidt laws.

In order to investigate gravitational instability in disk galaxiesand consequent star formation, we model isolated galaxies witha wide range of masses and gas fraction. In Paper I we havedescribed the galaxy models and computational methods and dis-cussed the star formation morphology associated with gravita-tional instability. In that paper it was shown that the nonlineardevelopment of gravitational instability determines where andwhen star formation takes place and that the star formation time-scale �SF depends exponentially on the initial Toomre instabilityparameter for the combination of collisionless stars and collisionalgas in the diskQsg derived by Rafikov (2001). Galaxies with highinitial mass or gas fraction have small Qsg and are more unstable,forming stars quickly, while stable galaxies with Qsg > 1 main-tain quiescent star formation over a long time.

Paper I emphasized that to form a stable disk and derive thecorrect SFR from a numerical model, the gravitational collapseof the gasmust be fully resolved (Bate &Burkert 1997, hereafterBB97; Truelove et al. 1997) up to the density where gravita-tionally collapsing gas decouples from the flow. If this is done,stable disks with SFRs comparable to observed values can bederived from models using an isothermal equation of state. Withinsufficient resolution, however, the disk tends to collapse to thecenter, producing much higher SFRs, as found by some previouswork (e.g., Robertson et al. 2004).

We analyze the relation between the SFR and the gas density,both globally and locally. In x 2 we briefly review our computa-tional method, galaxy models, and parameters. In x 3 we presentthe evolution of the SFR and radial distributions of both gas andstar formation.We derive the global Schmidt law in x 4, followedby a parameter study and an exploration of alternative forms ofthe star formation law. Local Schmidt laws are presented in x 5.In x 6 we investigate the SFE. The assumptions and limitationsof the models are discussed in x 7. Finally, we summarize ourwork in x 8. Preliminary results on the global Schmidt law andstar formation thresholds were already presented by Li et al.(2005a).

2. COMPUTATIONAL METHOD

We here summarize the algorithms, galaxy models, and nu-merical parameters described in detail in Paper I.We use the SPHcode GADGET, version 1.1 (Springel et al. 2001), modified toinclude absorbing sink particles (Bate et al. 1995) to directlymeasure the mass of gravitationally collapsing gas. Paper I andJappsen et al. (2005) give detailed descriptions of sink particleimplementation and interpretation. In short, a sink particle is cre-ated from the gravitationally bound region at the stagnation point

of a converging flow where number density exceeds values ofn ¼ 103 cm�3. It interacts gravitationally and inherits the massand linear and angular momentum of the gas. It accretes sur-rounding gas particles that pass within its accretion radius andare gravitationally bound.

Regions where sink particles form have pressures P/k �107 K cm�3 typical of massive star-forming regions.We interpretthe formation of sink particles as representing the formation ofmolecular gas and stellar clusters. Note that the only regions thatreach these high pressures in our simulations are dynamicallycollapsing. The measured mass of the collapsing gas is insen-sitive to the value of the cutoff density. This is not an importantfree parameter, unlike in the models of Elmegreen (2002) andKrumholz & McKee (2005).

Our galaxy model consists of a dark matter halo and a disk ofstars and isothermal gas. The initial galaxy structure is based onthe analytical work by Mo et al. (1998), as implemented numer-ically by Springel &White (1999), Springel (2000), and Springelet al. (2005).We characterize ourmodels by the rotational velocityV200 at the virial radius R200 where the density reaches 200 timesthe cosmic average. We have run models of galaxies with ro-tational velocity V200 ¼ 50 220 km s�1, with gas fractions of20%–90% of the disk mass for each velocity.

Observations of H i in many spiral galaxies suggest that thegas velocity dispersion has a range of �4–15 km s�1 (e.g., seereview in Dib et al. 2005). The dispersion �H i varies radiallyfrom �12–15 km s�1 in their central regions to�4–6 km s�1 inthe outer parts (e.g., van der Kruit & Shostak 1982; Dickey et al.1990; Kamphuis & Sancisi 1993; Rownd et al. 1994; Meureret al. 1996). We therefore choose two sets of effective soundspeeds for the gas: cs ¼ 6 km s�1 ( low-temperature models), assuggested by Kennicutt (1998a), and cs ¼ 15 km s�1 (high-temperature models). Table 1 lists the most important model pa-rameters. The Toomre criterion for gravitational instability thatcouples stars and gas,Qsg, is calculated following Rafikov (2001),and the minimum value is derived using the wavenumber k ofgreatest instability and lowest Qsg at each radius.

Models of gravitational collapse must satisfy three numer-ical criteria: the Jeans resolution criterion (BB97; Whitworth1998), the gravity-hydro balance criterion for gravitational soft-ening (BB97), and the equipartition criterion for particle masses(Steinmetz & White 1997). We set up our simulations to satisfythe above three numerical criteria, with the computational pa-rameters listed in Table 1. We choose the particle number foreach model such that they not only satisfy the criteria, but also sothat all runs have at least 106 total particles. The gas, halo, andstellar disk particles are distributed with number ratioNgNhNd ¼532. The gravitational softening lengths of the halo and disk arehh ¼ 0:4 kpc and hd ¼ 0:1 kpc, respectively, while that of thegas hg is given in Table 1 for each model. The minimum spatialand mass resolutions in the gas are given by hg and twice the ker-nel mass (�80mg). (Note that we use hg here to denote the grav-itational softening length instead of � as used in previous papers, todistinguish it from the SFE used in later sections.) We adopt typ-ical values for the halo concentration parameter c ¼ 5, spin pa-rameter k ¼ 0:05, and Hubble constant H0 ¼ 70 km s�1 Mpc�1

(Springel 2000).Resolution of the Jeans length is vital for simulations of

gravitational collapse (Truelove et al. 1997; BB97). Exactly howwell the Jeans length must be resolved remains a point of con-troversy, however. Truelove et al. (1997) suggest that a Jeansmass must be resolved with far more than the Nk ¼ 2 smoothingkernels proposed by BB97. In this work, we carried out a reso-lution study on low-T models G100-1 and G220-1, with three

STAR FORMATION IN ISOLATED DISK GALAXIES. II. 881No. 2, 2006

resolution levels having total particle numbers Ntot ¼ 105 (R1),8 ; 105 (R8), and 6:4 ; 106 (R64), reaching Nk � 24. The par-ticle numbers are chosen such that the maximum spatial resolu-tion increases by a factor of 2 between each pair of runs. Paper Ifinds convergence to within 10% of the global amount of massaccreted by sink particles between the two highest resolutions,suggesting that the BB97 criterion is sufficient for the problem ofglobal collapse in galactic disks. In this paper we also refer to theresolution study as applicable.

3. STAR FORMATION AND GAS DISTRIBUTION

Molecular clouds and stars form together in galaxies. Molec-ular hydrogen forms on dust grains in a time of (Hollenbach et al.1971)

tf ’ 109 yr� �

=n; ð2Þ

where n is the number density of the gas. The absence ofk10Myrold stars in star-forming regions in the solar neighborhood sug-gests that molecular cloud complexes must coalesce rapidly andform stars quickly (Ballesteros-Paredes et al. 1999; Hartmann2000). Hartmann et al. (2001) further suggested that the condi-tions needed for molecular gas formation from atomic flows aresimilar to the conditions needed for gravitational instability. Starformation can therefore take place within a free-fall timescaleonce molecular clouds are produced. Using a one-dimensionalchemical model, Bergin et al. (2004) showed rapid formation ofmolecular gas in 12–20Myr in shock-compressed regions. Theseresults are confirmed by S. Glover & M.-M. Mac Low (2006, in

preparation) using three-dimensionalmagnetohydrodynamics sim-ulations with chemistry of supersonic turbulence. They show thatmost of the atomic gas turns into H2 in just a few million yearsonce the average gas density rises above �100 cm�3. This is be-cause the gas passes through turbulent density fluctuations ofhigher densitywhereH2 can formquickly. By the time gas reachesthe densities of n ¼ 1000 cm�3 where we replace it with sink par-ticles, the molecular hydrogen formation timescale tf P 1 Myr(eq. [2]). Motivated by these results, we identify the high-densityregions formed by gravitational instability as giant molecularcloud complexes and replace these regions by accreting sinkparticles.We assume that a fraction of the molecular gas turns into stars

quickly. CO observations by Young et al. (1996) and Rownd &Young (1999) suggest that the local SFE in molecular cloudsremains roughly constant. To quantify the SFR, we thereforeassume that individual sink particles form stars at a fixed localefficiency �‘.Kennicutt (1998a) found a global SFE of �g ¼ 30% for star-

burst galaxies, which Wong & Blitz (2002) found to be domi-nated by molecular gas. We take this to be a measure of the localSFE �‘ in individual molecular clouds, since most gas in thesegalaxies has already become molecular. In our simulations, sinkparticles represent high-pressure (P/k � 107 K cm�3), massivestar formation regions in galaxies, such as 30 Doradus in theLarge Magellanic Cloud (e.g., Walborn et al. 1999). We there-fore adopt a fixed local SFE of �‘ ¼ 30% to convert the mass ofsink particles to stars, while making the simple approximationthat the remaining 70% of the sink particle mass remains inmolecular form. This approach is discussed in more detail in x 6.

TABLE 1

Galaxy Models and Numerical Parameters

Modela fgb

Rdc

( kpc) Qsg(LT)d Qsg(HT)

e

Ntotf

(106)

hgg

(pc)

mgh

(104 M�)

�SF (LT)i

(Gyr)

�SF (HT)j

(Gyr)

G50-1 ......................... 0.2 1.41 1.22 1.45 1.0 10 0.08 4.59 . . .

G50-2 ......................... 0.5 1.41 0.94 1.53 1.0 10 0.21 1.28 . . .

G50-3 ......................... 0.9 1.41 0.65 1.52 1.0 10 0.37 0.45 . . .G50-4 ......................... 0.9 1.07 0.33 0.82 1.0 10 0.75 0.15 0.53

G100-1 ....................... 0.2 2.81 1.08 1.27 6.4 7 0.10 2.66 . . .

G100-2 ....................... 0.5 2.81 . . . 1.07 1.0 10 1.65 . . . . . .G100-3 ....................... 0.9 2.81 . . . 0.82 1.0 10 2.97 . . . 1.92

G100-4 ....................... 0.9 2.14 . . . 0.42 1.0 20 5.94 . . . 0.15

G120-3 ....................... 0.9 3.38 . . . 0.68 1.0 20 5.17 . . . 0.46

G120-4 ....................... 0.9 2.57 . . . 0.35 1.0 30 10.3 . . . 0.16

G160-1 ....................... 0.2 4.51 . . . 1.34 1.0 20 2.72 . . . 3.1k

G160-2 ....................... 0.5 4.51 . . . 0.89 1.0 20 6.80 . . . 0.58

G160-3 ....................... 0.9 4.51 . . . 0.52 1.0 30 12.2 . . . 0.30

G160-4 ....................... 0.9 3.42 . . . 0.26 1.5 40 16.3 . . . 0.11

G220-1 ....................... 0.2 6.20 0.65 1.11 6.4 15 1.11 0.28 3.0k

G220-2 ....................... 0.5 6.20 . . . 0.66 1.2 30 14.8 . . . 0.39

G220-3 ....................... 0.9 6.20 . . . 0.38 2.0 40 15.9 . . . 0.25

G220-4 ....................... 0.9 4.71 . . . 0.19 4.0 40 16.0 . . . 0.096

a First number is rotational velocity in km s�1 at the virial radius, the second number indicates submodel. Submodels have varying fractions md of totalhalo mass in their disks and given values of fg. Submodels 1–3 have md ¼ 0:05, while submodel 4 has md ¼ 0:1.

b Fraction of disk mass in gas.c Stellar disk radial exponential scale length.d Minimum initial value of Qsg(R) for low-T models.e Minimum initial value of Qsg(R) for high-T models.f Total particle number.g Gravitational softening length of gas.h Gas particle mass.i Star formation timescale of low-T model (from Paper I).j Star formation timescale of high-T model (from Paper I).k Maximum simulation time step instead of the star formation timescale �SF.


3.1. Evolution of Star Formation Rate

Figure 1 shows the time evolution of the SFRs of differentmodels. The SFR is calculated as SFR ¼ dM�/dt ’ �‘�Msink/�t,whereM� andMsink are the masses of the stars and sink particles,respectively.We choose �‘ ¼ 30% to be the local SFEwithin sinkparticles and the time interval �t ¼ 50 Myr. The top panel ofFigure 1 shows the SFR curves of selected high-T models, whilethe bottom panel of Figure 1 shows a resolution study of SFRevolution.We find convergence to within 10%of the SFR betweenthe two highest resolutions over periods of more than 2 Gyr, sug-gesting that this result converges well under the BB97 criterion.

The SFRs in Figure 1 decline over time. Many of the modelshave SFR > 10 M� yr�1, corresponding to starburst galaxies.Some small or gas-poor models such as G100-1, on the otherhand, have SFR < 1M� yr�1. They maintain slow but steady starformation over a long time and may represent quiescent normalgalaxies. The maximum SFR appears to depend quantitatively onthe initial instability of the disk as measured either by Qsg;min,the minimum Toomre parameter for the combination of starsand gas in the disk, or the value for the gas only Qg;min. Figure 2shows both correlations: log SFR max ’ 3:32� 3:13Qsg;min andlog SFR max ’ 2:76� 1:99Qg;min.

The SFRs in most models shown in Figure 1 appear to declineexponentially. From Paper I, the accumulated mass of the starsformed in each galaxy can be fitted with an exponential function:

M�

Minit

¼ M0 1� exp �t=�SFð Þ½ �; ð3Þ

where

M0 ¼ 0:96�‘ 1� 2:9 exp �1:7=Qsg;min

� �� ; ð4Þ

�SF ¼ 34 � 7 Myrð Þ exp Qsg;min=0:24� �

; ð5Þ

Minit is the initial total gas mass, and �SF is the star formationtimescale. The SFR can then be rewritten in the following form:

SFR ¼ dM�

dt/ 1

�SFexp �t=�SFð Þ: ð6Þ

A similar exponential form is also reported by MacArthur et al.(2004), as first suggested by Larson (1974) and Tinsley &Larson (1978).

Sandage (1986) studied the SFR of different types of galaxiesin the Local Group and proposed an alternative form for the starformation history, as explicitly formulated by MacArthur et al.(2004):

SFR / t

� 2SF

exp �t2=� 2SF

� �: ð7Þ

The top panel of Figure 3 shows an example of model G220-1(low T ) fitted with these two formulae. Both formulae appearquite similar at intermediate times. The Sandage model capturesthe initial rise in star formation better, but the exponential formfollows the late-time behavior of our models more closely. As weonly include stellar feedback implicitly by maintaining constantgas sound speed, we must be somewhat cautious about our in-terpretation of the late-time results. In order to compare the fits,we define a parameter for relative goodness of the fit

�2 ¼X

ys � yf� �

=ym� �2

; ð8Þ

where ys is the SFR from the simulation, ym is the maximumof the SFR, and yf is the model function from equation (6) orequation (7). Note that since we do not take into account theuncertainty of each point, the absolute value of �2 has nomeaning. We only compare the relative �2 values in the bottompanel of Figure 3. Both formulae fit equally well to many mod-els, especially to those with high gas fractions that form a lot ofstars early on. But for somemodels such as G100-1 ( low T ) andG220-1 (high T ), the exponential function seems to fit notice-ably better. Therefore, we use the exponential form in the rest ofthe paper.

Fig. 1.—Top: Time evolution of the SFRs in selected high-Tmodels as givenin the legend. Bottom: Resolution study of low-T models G100-1 (blue) andG220-1 (orange) with resolutions of R1 (dotted lines), R8 (dashed lines), andR64 (solid colored lines), where the resolution levels are in units of 105 totalparticles. The standard R10 models (black) are shown for comparison.


This analysis implies that the star formation history dependsquantitatively on the initial gravitational instability of a galaxyafter its formation or any major perturbation. An unstable galaxyforms stars rapidly in an early time, so its stellar populations willappear older than those in a more stable galaxy. More massivegalaxies are less stable than small galaxies with the same gasfraction. The different star formation histories in such galaxiesmay account for the downsizing effect that star formation first

occurs in big galaxies at high redshift, while modern starburstgalaxies are smaller (Cowie et al. 1996; Poggianti et al. 2004;Ferreras et al. 2004) and thus more stable.

3.2. Radial Distribution of Gas and Star Formation

Figure 4 shows the radial distribution of different gas com-ponents and the SFR of selected models. The gas distributionand SFR are calculated at the star formation timescale �SF de-rived from the fits given in Paper I. We assume that 70% of the

Fig. 3.—Top: Example fit of the SFR evolution curve with the exponentialform (eq. [6]) and the Sandage (1986) form (eq. [7]) for model G220-1 ( low T )as an example. Bottom: Relative goodness of fit �2 (eq. [8]) for all models forexponential (red ) and Sandage (1986) (black) forms.

Fig. 2.—Correlation between the maximum of the SFR and the minimumradial values of the Toomre instability parameters for stars and gas togetherQsg;min (top) and for gas alone Qg;min (bottom). The solid lines are the least-absolute-deviation fits log SFRmax ¼ K þ AQmin.


gravitationally collapsed, high-density gas (as identified by sinkparticles) is in molecular form. Similarly, we identify unaccretedgas as being in atomic form. The total amount of gas is the sum ofboth components.

Figure 4 shows that the simulated disks have gas distributionsthat are mostly atomic in the outer disk but dominated by mo-lecular gas in the central region. Observations by Wong & Blitz(2002) and Heyer et al. (2004) show that the density differencebetween the atomic and molecular components in the centralregion depends on the size and gas fraction of the galaxy. Forexample, �H2

in the center of NGC 4321 is almost 2 orders ofmagnitude higher than�H i (Wong & Blitz 2002), while in M33,the difference is only about 1 order of magnitude (Heyer et al.

2004). A similar relation between the fraction of molecular gasand the gravitational instability of the galaxy is seen in our sim-ulations. In our most unstable galaxies such as G220-4, the cen-tral molecular gas surface density exceeds the atomic gas surfacedensity by more than 2 orders of magnitude, while in a morestable model like G220-1 (high T ), the profiles of �H2

and �H i

are close to each other within one disk scale length Rd.We also find a linear correlation between the molecular gas

surface density and the SFR surface density, as can be seen bytheir parallel radial profiles in Figure 4. (In operational terms, wefind a correlation between the surface density in sink particles andthe rate at which they accrete mass.) Gao & Solomon (2004a)found a tight linear correlation between the far-infrared luminosity,

Fig. 4.—Radial profiles of atomic (SPH particles; green diamonds), molecular (70% of sink particle mass; blue squares), and total (red circles) gas surfacedensity, as well as SFR surface density (black circles). Rd is the radial disk scale length as given in Table 1. Note that the solid lines are used to connect the symbols.


a tracer of the SFR, and HCN luminosity, in agreement with ourresult that SFR and molecular gas have similar surface densityprofiles.

The agreement between the simulations and observationssupports our assumption that both molecular gas and stars formby the gravitational collapse of high-density gas. Note, however,that we neglect recycling of gas frommolecular clouds back intothe warm atomic and dissociated or ionized medium representedby SPH particles in our simulation. Although it is possible thateven that reionized gas may still quickly collapse again if theentire region is gravitationally unstable, this still constitutes animportant limitation of our models that will have to be addressedin future work.

4. GLOBAL SCHMIDT LAW

To derive the global Schmidt law from our models, we aver-age �SFR and �gas over the entire star-forming region of eachgalaxy. We define the star-forming region following Kennicutt(1989), using a radius chosen to encircle 80% of the mass ac-cumulated in sink particles (denoted R80 hereafter). The SFR istaken from the SFR evolution curves at some chosen time. Asmentioned in x 3, 30% of the mass of sink particles is assumedto be stars, while the remaining 70% of the sink particle massremains in molecular form. The atomic gas component is com-puted from the SPH particles not participating in localized gravi-tational collapse, that is, gas particles not accreted onto sinkparticles. The total is the sum of atomic and molecular gas.

Figure 5 shows the global Schmidt laws derived from oursimulations at the star formation time t ¼ �SF as listed in Table 1.Note that for a few models this is just the maximum simulatedtime, as indicated in Table 1. (Results from different times andstar formation regions are shown in the next section.) We fit thedata to the total gas surface density of themodels listed in Table 1(both low T and high T ). A least-squares fit to the models wehave run gives a simulated global Schmidt law

�SFR ¼ 1:1 � 0:4ð Þ ; 10�4 M� yr�1 kpc�2� �

;�gas

1 M� pc�2

� �1:56�0:09

: ð9Þ

For comparison, the best fit to the observations by Kennicutt(1998a) gives a global Schmidt law for the total gas surfacedensity in a sample that includes both the normal and starburstgalaxies of

�SFR ¼ 2:5 � 0:7ð Þ ; 10�4 M� yr�1 kpc�2� �

;�gas

1 M� pc�2

� �1:4�0:15

: ð10Þ

The global Schmidt law derived from the simulations agreeswith the observed slope within the observational errors but has anormalization a bit lower than the observed range. There arethree potential explanations for this discrepancy. First, we havenot weighted the fit by the actual distribution of galaxies in massand gas fraction. Second, as we discuss in the next subsection,we have not used models at different times in their lives weightedby the distribution of lifetimes currently observed. Third, we haveonly simulated isolated, normal galaxies. Our models thereforedo not populate the highest �SFR values observed in Kennicutt(1998a), which are all starbursts occurring in interacting galaxies.These produce highly unstable disks that undergo vigorous star-

bursts with high SFR (e.g., Li et al. 2004). Our result is supportedby Boissier et al. (2003), who found a much deeper slope, N � 2,in a sample of normal galaxies comparable to our more stablemodels. In the models, the local SFE �‘ is fixed at 30%, inde-pendent of the galaxy model. A change of the assumed value of �‘changes the normalization but not the slope of our relation. Forexample, an extremely high value of �‘ � 90% increases A to�3:15 ; 10�4 M� yr�1 kpc�2, which is just within the 1 � upperlimit of the observation by Kennicutt (1998a). If we decrease �‘ to10%, then A decreases to �0:7 ; 10�4 M� yr�1 kpc�2. Thesefairly extreme assumptions still produce results lying within theobserved ranges (e.g., Wong & Blitz 2002), suggesting that ouroverall results are insensitive to the exact value of the local SFEthat we assume.The SFR surface densities �SFR change dramatically with the

gas fraction in the disk. The most gas-rich models (M-4; circles)have the highest �SFR, while the models poorer in gas (M-1;squares) have �SFR 2 orders of magnitudes lower than their gas-rich counterparts. Note that models with lower �SFR tend to haveslightly higher scatter because in thesemodels fewer sink particlesform, and they form over a longer period of time, resulting inhigher statistical fluctuations.A resolution study is shown in Figure 6 that compares the

global Schmidt law computed with different numerical resolu-tions. Runs with different resolution converge within 10% in boththe �SFR and �gas. Although numerical resolution affects the to-tal mass collapsed and the number and location of fragments, as

Fig. 5.—Comparison of the global Schmidt laws between our simulations andthe observations. The red line is the least-squares fit to the total gas of the simulatedmodels, the black solid line is the best fit of observations from Kennicutt (1998a),while the black dotted lines indicate the observational uncertainty. The color of thesymbol indicates the rotational velocity for each model (see Table 1), labels fromM-1 to M-4 are submodels with increasing gas fraction, and open and filledsymbols represent low- and high-T models, respectively.


shown in Paper I, the SFR at �SF seems to be less sensitive to thenumerical resolution.

4.1. A Parameter Study

In order to test how sensitive the global Schmidt law is to theradius R and the time t chosen to measure it, we carry out aparameter study changing both R and t individually. To maintainconsistency with the previous section, we continue to assumea constant local SFE �‘ ¼ 30%. Figure 7 compares the globalSchmidt laws in total gas at different radii for the star-formingregion R ¼ R50 and R100 (encircling 50% and 100% of the newlyformed star clusters), while the time is fixed to t ¼ �SF. We cansee that the case with R50 has larger scatter than that with R100.This is due to the larger statistical fluctuations caused by thesmaller number of star clusters within this radius. The globalSchmidt law with R ¼ R100 is almost identical to that withR ¼ R80 shown in Figure 5.

Figure 8 compares the global Schmidt laws in total gas atdifferent times t ¼ 0:5�SF and 1.5�SF with the star formationradius fixed to R80. Compared to the t ¼ �SF case, models in thet ¼ 0:5�SF case have higher �SFR because the SFR drops almostexponentially with time (x 3.1). Similarly, models in the t ¼1:5�SF case shift to the lower right. Nevertheless, data derivedfrom different times appear to preserve the power-law index ofthe Schmidt law, just differing in the normalizations.

The global Schmidt law presented by Li et al. (2005a) wasderived at a time when the total mass of the star clusters reached70% of the maximum collapsed mass, which is close to 1.0�SFin many models. The time interval�t used to calculate the SFRwas the time taken to grow from 30% to 70% of the maximum

Fig. 6.—Same as Fig. 5, but for the resolution study of low-Tmodels of G100-1(blue) and G220-1 (orange). Models with total particle number of Ntot ¼ 105 (R1;open triangle), 8 ; 105 (R8; open circle), and 6:4 ; 106 (R64; open square) areshown. Models with regular resolution Ntot ¼ 106 (R10; filled circle) are alsoshown for comparison.

Fig. 7.—Same as Fig. 5, but with different radii assumed for the star-formingregion: R ¼ R50 (top) and R ¼ R100 (bottom). The time is fixed at t ¼ �SF.


collapsed mass, rather than the �t ¼ 50 Myr used here. Never-theless, the results presented here also agree well with those in Liet al. (2005a).Our parameter study demonstrates that the global Schmidt law

depends only weakly on the details of how it is measured. Thesmall scatter seen in Kennicutt (1998a) does suggest that addi-tional physics not included in our modeling may be important.We should keep in mind that since we do not treat gas recycling,our models are valid only within one gas consumption time �SF.The evolution after that may become unrealistic as most of thegas is locked up in the sinks. Nevertheless, our results suggestthat the Schmidt law is a universal description of gravitationalcollapse in galactic disks.

4.2. Alternative Global Star Formation Laws

The existence of a well-defined global Schmidt law suggeststhat the SFR depends primarily on the gas surface density.As shown by several authors (e.g., Quirk 1972; Larson 1988;Kennicutt 1989, 1998a; Elmegreen 1994), a simple picture ofgravitational collapse on a free-fall timescale �A / ��1/2 qual-itatively produces the Schmidt law. Assuming that the gas sur-face density is directly proportional to the midplane density,�gas / �, it follows that �SFR / �gas/�A / �3/2

gas. This suggeststhat the Schmidt law reflects the global growth rate of gas densityunder gravitational perturbations.An alternative scenario that uses the local dynamical time-

scale has been suggested by several groups (e.g., Wyse 1986;Wyse & Silk 1989; Silk 1997; Elmegreen 1997; Hunter et al.1998; Tan 2000). In particular, Elmegreen (1997) and Hunteret al. (1998) proposed a kinematic law that accounts for thestabilizing effect of rotational shear, in which the global SFRscales with the angular velocity of the disk,

�SFR / �gas

torb/ �gas�; ð11Þ

where torb is the local orbital timescale and � is the orbital fre-quency. Kennicutt (1998a) gave a simple form for the kine-matical law,

�SFR ’ 0:017�gas�; ð12Þ

with the normalization corresponding to an SFR of 21% of thegas mass per orbit at the outer edge of the disk.For our analysis, we follow Kennicutt (1998a) and define

torb ¼ 2�R /V (R) ¼ 2�/�(R), where V(R) is the rotational veloc-ity at radius R. We use the initial rotational velocity, which shouldnot change much with time as it depends largely on the potentialof the dark matter halo. Figure 9 shows the relationship between�SFR and �gas� in our models. The densities of SFR �SFR andtotal gas�gas are calculated the same way as in Figure 5 at 1.0�SFand R80, and�(R) is calculated by using the initial total rotationalvelocity at R80. A least-squares fit to the data gives �SFR ¼(0:036 � 0:004)(�gas�)1:49�0:1. This correlation has a steeperslope than the linear law given in equation (12), suggesting adiscrepancy between the behavior of our models and the observedkinematical law.However, Boissier et al. (2003) recently reported a slope of

�1.5 for the kinematical law from observations of 16 normaldisk galaxies, in agreement with our results. Examination of Fig-ure 7 in Kennicutt (1998a) also shows that the normal galaxies,considered alone, seem to have a steeper slope than the galacticnuclei and starburst galaxies. Boissier et al. (2003) suggested sev-eral reasons for the discrepancy, the most important one being the

Fig. 8.—Same as Fig. 5, but derived at different times t ¼ 0:5�SF (top) andt ¼ 1:5�SF (bottom). The radius for the star formation region is fixed at R80.


difference between their sample of normal galaxies and the sampleof Kennicutt (1998a) including many starburst galaxies and gal-axy nuclei. More simulations, as well as models with higher SFRsuch as galaxy mergers, are necessary to test this hypothesis.

4.3. A New Parameterization

The global Schmidt law describes global collapse in the gasdisk. It does not seem to depend on the local star formationprocess. From Paper I and Li et al. (2005a) we know that �SFR

correlates tightly with the strength of gravitational instability(see alsoMac Low&Klessen 2004; Klessen et al. 2000; Heitschet al. 2001). Here we quantify this correlation, using the ToomreQ parameter to measure the strength of instability.

Figure 10 shows the correlation between �SFR and the localgravitational instability parametersQmin. The parametersQmin areminimumvalues of the ToomreQ parameters for gasQg;min(t) andthe combination of stars and gas Qsg;min(t) at a given time t. Toobtain the Q parameters, we follow the approach of Rafikov(2001), as described in equations (1)–(3) of Paper I.We divide theentire galaxy disk at time t into 40 annuli, calculate the Q pa-rameters in each annulus, and then take theminimum. In the plots,the time when �SFR is computed is t ¼ �SF. This correlation doesnot change significantly with time, but the scatter becomes largerat later times because the disk becomes more clumpy, whichmakes the calculation of Qg;min(t) more difficult (see below).

There is substantial scatter in the plots, at least partly caused bythe clumpy distribution of the gas. Equations (1)–(3) for the Qparameters in Paper I are derived for uniformly distributed gas,such as in our initial conditions. As the galaxies evolve, the gas

forms filaments or spiral arms probably leading to the fluctuationsseen. The least-squares fits to the data shown in Figure 10 give

�SFR ¼ 0:013 � 0:003 M� yr�1 kpc�2� �

Qsg;min(�SF)� ��1:54�0:23

ð13Þ

¼ 0:019 � 0:005M� yr�1 kpc�2� �

Qg;min(�SF)� ��1:12�0:21

:

ð14Þ

Fig. 9.—Relation between �SFR and �gas�. The legends are the same as inFig. 5. The black solid line is the linear relation derived from the observationsby Kennicutt (1998a), as given in our equation (12).

Fig. 10.—Correlations between �SFR and the minimum value of the Toomreparameter for stars and gasQsg(t) (top) and the Toomre parameter for gasQg(t) att ¼ 1�SF (bottom). The legends are the same as in Fig. 5. The solid black linesare the least-squares fits to the data given in equation (13).


If we take a first-order approximation, Qsg / 1/�gas, thenequation (13) gives �SFR / �1:54

gas at t ¼ �SF, agreeing very wellwith the observations. The slopes derived from Qg appear to belower than those derived from Qsg but are still within the sloperange observed.

Keep in mind that the local instability is a nonlinear interac-tion between the stars and gas and so is much more complicatedthan the linear stability analysis presented here. Also, the in-stability of the entire disk at a certain time is not fully representedby the minimum values of the Q parameters we employ here,although they do represent the region of fastest star formation.These factors limit our ability to derive the global Schmidt lawdirectly from the instability analysis.

5. LOCAL SCHMIDT LAW

The relationship between surface density of SFR�SFR and gasdensity �gas can also be measured as a function of radius withina galaxy, giving a local Schmidt law. Observations by Wong &Blitz (2002) and Heyer et al. (2004) show significant variationsin both the indices N and normalizations A of the local Schmidtlaws of individual galaxies. For example, Heyer et al. (2004)show that M33 follows the law

�SFR ¼ 0:0035 � 0:066 M� yr�1 kpc�2� �; �tot=1 M� pc�2� �3:3�0:07

; ð15Þ

while Wong & Blitz (2002) show that N has a range of 1.23–2.06 in their sample.

To derive local Schmidt laws, we again divide each individualgalaxy into 40 radial annuli within 4Rd and then compute �gas

and �SFR in each annulus. The SFR is measured at t ¼ �SF as inx 4. For models where �SF > 3 Gyr or is beyond the simulationduration, the maximum simulated time step is used instead, aslisted in Table 1.

5.1. Star Formation Thresholds

Figure 11 shows the relation between �SFR and �gas correla-tions for all models in the simulations that form stars in the first3 Gyr. With such a large number of models in one plot, it isstraightforward to characterize the general features and to com-pare with the observations shown in Figure 3 of Kennicutt(1998a). Similar to the individual galaxies in Kennicutt (1998a)and Martin & Kennicutt (2001), each model here shows a tight�SFR-�gas correlation, the local Schmidt law. However, �SFR

drops dramatically at some gas surface density. This is a clearindication of a star formation threshold.

We therefore define a threshold radius Rth as the radius that en-circles 95% of the newly formed stars. The gas surface density atthe threshold radiusRth in Figure 11 has a range from�4M� pc�2

for the relatively stable model G220-1 (low T ) to �60 M� pc�2

for the most unstable model G220-4 (high T ). Note that in somegalaxies, there are also smaller dips in SFR at higher density.Martin & Kennicutt (2001) suggest that rotational shearing cancause an inner star formation threshold. However, the inner dips inour simulations are likely due to the lack of accretion onto sinkparticles in the simulations after most of the gas in the central re-gion has been consumed. Further central star formation in realgalaxies would occur due to gas recycling, which we neglect, and,probably more important, after interactions with other galaxies.

In the analysis of observations, a dimensionless parameter,�Q ¼ �th /�crit ¼ 1/Q, has been introduced to relate the starformation threshold to the Toomre unstable radius (Kennicutt1989). The critical radius is usually defined as the radius where

Qg ¼ 1. With a sample of 15 spiral galaxies, Kennicutt (1989)found �Q ’ 0:63 by assuming a constant effective sound speed(the velocity dispersion) of the gas cs ¼ 6 km s�1. This resultwas confirmed by Martin & Kennicutt (2001) with a larger sam-ple of 32 well-studied nearby spiral galaxies, who reported arange of �Q � 0:3 1:2, with a median value of 0.69. However,Hunter et al. (1998) found �Q ’ 0:25 for a sample of irregulargalaxies with cs ¼ 9 km s�1. As pointed out by Schaye (2004),this derivation of�Q depends on the assumption of cs. The valuesof �Q derived from our models using their actual values of cs areshown in Figure 12. We find that the value of �Q depends notonly on the gas sound speed but also on the gas fraction of thegalaxy. For models with the same rotational velocity and gasfraction, lower gas sound speed results in a higher value of �Q.For models with the same total mass and sound speed, higher gasfraction leads to higher �Q. The gas-poor models in our simu-lations ( fg ¼ 20%) have a range of �Q � 0:2 1:0, agreeingroughly with observations. This again may reflect the relativestability of the nearby galaxies in the observed samples.There are several theoretical approaches to explain the presence

of star formation thresholds. Martin & Kennicutt (2001) suggestthat the gravitational instability model explains the thresholdswell, with the deviation of �Q from one simply due to the nonuni-form distribution of gas in real disk galaxies. Hunter et al. (1998)proposed a shear criterion for star-forming dwarf irregular galax-ies, as they appear to be subcritical to the Toomre criterion. Schaye(2004) modeled the thermal and ionization structure of a gaseousdisk. He found that the critical density is about �crit � 3 10M�pc�2 with a gas velocity dispersion of�10 km s�1 and argued that

Fig. 11.—Local Schmidt laws of all models with �SF < 3 Gyr. The legends arethe same as in Fig. 5: the color of the symbol indicates the rotational velocity for eachmodel as given in Table 1, the shape indicates the submodel classified by gas frac-tion, and open and filled symbols represent low- and high-T models, respectively.


thermal instability determines the star formation threshold in theouter disk. Our models suggest that the threshold depends on thegravitational instability of the disk. The derived�crit and �Q fromour stable models (Qsg;min > 1) agree well with observations,supporting the arguments of Martin & Kennicutt (2001).

5.2. Local Correlations between Gas and Star Formation Rate

We fit the local Schmidt law to the total gas surface densitywithin Rth, as demonstrated in Figure 13. The models in Fig-ure 13 all have the same rotational velocity of 220 km s�1 butdifferent gas fractions and sound speeds. The local Schmidt lawsof these models vary only slightly in slope and normalization.

The top and bottom panels of Figure 14 compare the slope Nand normalization A of the local Schmidt laws for all models inTable 1 that form stars in the first 3 Gyr. The slope of the fit to thetotal gas in the top panel of Figure 14 varies from about 1.2 to1.7.Larger galaxies tend to have larger N. However, the averageslope is around 1.3, agreeing reasonably well with that of theglobal Schmidt law.

There is substantial fluctuation in the normalization of fits tothe total gas, as shown in the bottom panel of Figure 14. Thevariation is more than an order of magnitude, with gas-rich mod-els tending to have highA. However, the average value ofA settlesaround 2.2, agreeing surprisingly well with that of the globalSchmidt law. Overall, the averaged local Schmidt law gives

�SFR ¼ 2:46 � 1:62ð Þ ; 10�4 M� yr�1 kpc�2� �

;�tot

1 M� pc�2

� �1:31�0:15

: ð16Þ

The averaged local Schmidt law is very close to the globalSchmidt law in x 4.

The average slope in equation (16) is rather smaller than thevalue of N ¼ 3:3 observed by Heyer et al. (2004) in M33. Thegalaxy M33 is very interesting. It is a nearly isolated, small diskgalaxy with low luminosity and low mass. It has total mass ofMtot � 1011 M� and a gas mass of Mgas � 8:0 ; 109 M� (Heyeret al. 2004). It is molecule-poor and subcritical, with gas sur-face density much smaller than the threshold surface density forstar formation found by Martin & Kennicutt (2001). However,it is actively forming stars (Heyer et al. 2004). We do not havea model that exactly resembles M33, although a close one mightbe model G100-1 in terms of mass. However, the gas veloc-ity dispersion of M33 is unknown, so we cannot make a directcomparison with our G100-1 models. In Figure 14, the low-Tmodel G100-1 has N � 1:4, but we have not derived a valuefor its high-T counterpart, as it does not form stars at all in thefirst 3 Gyr. Any stars that form in a disk similar to this will likelyform in spiral arms or other nonlinear density perturbationsthat are not well characterized by an azimuthally averaged sta-bility analysis. If these perturbations occur in the highest sur-face density regions as might be expected, the local Schmidt lawwill have a very high slope as observed. This speculation willneed to be confirmed with models reaching higher mass reso-lution in the future. The details of the feedback model andequation of state may also begin to play a role in this extremecase.

The averaged values of our derived local Schmidt laws doagree well with the observations by Wong & Blitz (2002)of a number of other nearby galaxies. The similarity betweenthe global and local Schmidt laws suggests a common origin

Fig. 12.—Dimensionless parameter giving the ratio of star formation thresh-old surface density to critical surface density for Toomre instability �Q ¼ 1/Qth

shown for low-T (red ) and high-T (black) models. Open symbols give valuesderived using gas only, �g ¼ 1/Qg;th, while filled symbols give values derivedusing both stars and gas, �sg ¼ 1/Qsg;th. Gas fraction of disks increases fromM-1to M-4 (see Table 1).

Fig. 13.—Comparisons of local Schmidt laws of galaxy models with thesame rotational velocity of 220 km s�1 but different gas fractions and soundspeeds, as indicated in the legend. The solid lines are least-squares fits to datapoints within the threshold radius Rth.


of the correlation between �SFR and �gas in gravitationalinstability.

6. STAR FORMATION EFFICIENCY

The SFE is poorly understood because it is difficult in bothobservations and simulations to determine the timescale for gasremoval and the gaseous and stellar mass within the star forma-tion region. On the molecular cloud scale, observations of sev-eral nearby embedded clusters with massM < 1000M� indicatethat the SFEs range from approximately 10% to 30% (Lada &

Lada 2003). However, it is thought that field stars form with SFEof only 1%–5% in giant molecular clouds (e.g., Duerr et al.1982), while the formation of a bound stellar cluster requires alocal SFEk 20% 50% (e.g., Wilking & Lada 1983; Elmegreen& Efremov 1997). An analytical model including outflows byMatzner & McKee (2000) suggests that the efficiency of clusterformation is in the range of 30%–50% and that of single starformation could be anywhere in the range of 25%–70%.In the analysis of our simulations presented here, we convert

the mass of the sink particles into stars using a fixed local SFE�‘ ¼ 30%, consistent with both the observations and theoreticalpredictions mentioned above. This local efficiency is differentfrom the global SFE in galaxies �g � �‘, which measures thefraction of the total gas turned into stars. On a galactic scale, theSFE appears to be associated with the fraction of molecular gas(e.g., Rownd & Young 1999). The global SFE has had valuesderived from observations over a wide range, depending on thegas distribution and the molecular gas fraction (Kennicutt 1998a).For example, in normal galaxies �g ’ 2% 10%, while in star-burst galaxies �g ¼ 10% 50%, with a median value of 30%. Onefactor that appears to contribute to the differences in �g is the gascontent. The global SFE is generally averaged over all gas com-ponents, but since star formation correlates tightly with the localgravitational instability, one expects higher global SFE in moreunstable galaxies. In fact, as pointed out byWong & Blitz (2002),most normal galaxies in the sample of Kennicutt (1998a) aremolecule-poor galaxies, which seem to have high stability andlow SFE, while molecule-rich starburst galaxies appear to be un-stable, forming stars with high efficiency.The variation of the normalization of the local Schmidt laws in

x 5 also suggests that the global SFE varies from galaxy togalaxy. To quantitatively measure the SFE in our models, weapply the common definition of the global SFE,

�g ¼ M�= M� þMgas

� �¼ M�=Minit; ð17Þ

over a period of 108 yr, an average timescale for star formationin a galaxy. In this equation,M� ¼ �‘Msink is the mass of newlyformed stars, and Mgas includes both the remaining mass of thesink particles and the SPH particles, so that M� þMgas ¼ Minit

is the total mass of the initial gas.The top and bottom panels of Figure 15 show the relation

between theminimumvalues of the initialQ parameterQsg;min andthe global SFE normalized by the local SFE �g /�‘. The time periodis taken as the first 100Myr after star formation starts. If we take �‘as a constant for all models, it appears that �g declines as Qsg;min

increases. Therefore, �g is high in less stable galaxies with highmass or high gas fraction. A least-absolute-deviation fit of the datagives a linear fit of �g /�‘ ¼ 0:9� 0:97Qsg;min. This fit is good forvalues ofQsg;min � 1. For more stable galaxies, with larger valuesof Qsg;min, the SFE remains finite, deviating from the linear fit.Using the empirical relations we have derived from our mod-

els earlier in the paper, we can derive a better analytic expres-sion for �g. Equation (17) can be combined with equation (3) inx 3.1 to yield an equation for the global SFE

�g ¼ M0 1� exp �t=�SFð Þ½ �: ð18Þ

We evaluate this at t ¼ 100 Myr, taking the definitions ofM0

and �SF derived from equations (4) and (5). Normalizing by thelocal SFE, we find

�g=�‘ ¼ 0:96 1� 2:88 exp �1:7=Qsg;min

� �� ; 1� exp �2:9e�Qsg;min=0:24

� �h i: ð19Þ

Fig. 14.—Local Schmidt law slope N (top) and normalization A (bottom) forall models with �SF < 3 Gyr. The fit is to the total gas surface density�tot withinthe threshold radius Rth. The symbols represent different models, as indicated inthe legends. The dotted line in each panel indicates the linearly averaged valueacross the models shown.


The function given by equation (19) is shown in Figure 15.For Qsg;min � 1:0 it is well approximated by the much simplerlinear function

�g=�‘ ’ 0:9� Qsg;min ð20Þ

as shown in the top panel of Figure 15. At larger values ofQsg;min,the exact function predicts the SFE in our models excellently as

shown in the bottom panel of Figure 15. Observational verifica-tion of this behavior is vital.

7. ASSUMPTIONS AND LIMITATIONS OF THE MODELS

7.1. Isothermal Equation of State

One of the two central assumptions in our model is the use ofan isothermal equation of state to represent a constant velocitydispersion. This is of course a simplification, as the interstellarmedium in reality has a broad range of temperatures 10 K <T < 107 K. However, neutral gas velocity dispersions in nor-mal spiral galaxies cover a far more limited range, as reviewedby Kennicutt (1998a), Elmegreen & Scalo (2004), Scalo &Elmegreen (2004), and Dib et al. (2005). The characteristic �increases when the averaged �SFR of a galaxy reaches tens ofsolar masses per year, but the normal galaxies in the sample withreliable measurements lie in the range �7–13 km s�1 (e.g.,Elmegreen & Elmegreen 1984; Meurer et al. 1996; van Zee et al.1997; Stil & Israel 2002; Hippelein et al. 2003).

At least twomechanisms appear viable for maintaining roughlyconstant velocity dispersion for the bulk of the gas in a galacticdisk: supernova feedback and magnetorotational instability (MacLow & Klessen 2004). Three-dimensional simulations in a peri-odic box with parameters characteristic of the outer parts of ga-lactic disks byDib et al. (2005) show that supernova driving leadsto constant velocity dispersions of � � 6 km s�1 for the total gasand �H i � 3 km s�1 for the H i gas, independent of the supernovarate. Simulations of the feedback effects across whole galacticdisks do suggest that the inner parts have slightly higher velocitydispersions (e.g., Thacker & Couchman 2000), although withinthe range that we consider. The magnetorotational instability ingalactic disks was suggested by Sellwood & Balbus (1999) tomaintain the observed velocity dispersion, a suggestion that hassince been substantiated by both local (Piontek & Ostriker 2005)and global (Dziourkevitch et al. 2004) numerical models. Thismay act even in regions with little or no active star formation.

Recently, Robertson et al. (2004) presented simulations ofgalactic disks and claimed that an isothermal equation of stateleads to a collapsed disk as the gas fragments into clumps thatfall to the galactic center due to dynamical friction. However,similar behavior is seen in models by Immeli et al. (2004), whodid not use an isothermal equation of state but also ran the sim-ulations at resolutions not satisfying the Jeans criterion (BB97;Truelove et al. 1997). On the other hand, using essentially thesame code and galaxy model as Robertson et al. (2004), but withhigher resolution satisfying the Jeans criterion, we do not seethis collapse. Insufficient resolution that fails to resolve the Jeansmass leads to spurious, artificial fragmentation and thus collapse.

Simulations by Governato et al. (2004) suggest that somelong-standing problems in galaxy formation such as the compactdisk and lack of angular momentum may well be due to insuf-ficient resolution or violation of numerical criteria. Our resultslead us to agree that the isothermal equation of state is not thecause of the compact disk problem, but rather inadequate nu-merical resolution.

Our assumption of an isothermal equation of state does, ofcourse, rule out the treatment by our model of phenomena suchas galactic winds associated with the hot phase of the interstellarmedium (although the venting of supernova energy verticallymay helpmaintain the isothermal behavior of the gas in the plane).The strong starbursts produced in some of our galaxy models willcertainly cause strong galactic winds. It remains unclear whethereven strong starbursts can remove substantial amounts of gas,however. Certainly they cannot in small galaxies (Mac Low &

Fig. 15.—Global SFE normalized by the local one �g /�‘ vs. the minimuminitial gravitational instability parameter Qsg;min in linear (top) and log space(bottom). The legends are the same as in Fig. 5. The black solid line is the least-absolute-deviation fit to the data, while the dotted line is the function given inequation (19).


Ferrara 1999), and larger galaxies would seem more resistant tostripping in starbursts than smaller ones. However, galactic windswill certainly influence the surroundings of starburst galaxies, aswell as their observable properties. These effects should eventu-ally be addressed in future simulations with more comprehensivegas physics and a more realistic description of the feedback fromstar formation.

7.2. Sink Particles

The use of sink particles enables us to directly identify highgas density regions, measure gravitational collapse, and followthe dynamical evolution of the system to a long time. We cantherefore determine the star formation morphologies and ratesand study the Schmidt laws and star formation thresholds.

However, one shortcoming of our sink particle implementa-tion is that we do not include gas recycling. Once the gas col-lapses into the sinks, it remains locked up there. As discussed inPaper I, the bulk of the gas that does not form stars will remain inthe disk and contribute to the next cycle of star formation. Also,the ejected material from massive stars will return into the gasreservoir for future star formation. Another problem is accretion.In the current model, sink particles accrete until the surroundinggas is completely consumed. However, in real star clusters, theaccretion would be cut off due to stellar radiation, and the clus-ters will actually lose mass due to outflow and tidal stripping.

These shortcomings of our sink particle technique may contrib-ute to two limitations of our models: first, the decline of SFRover time due to consumption of the gas, as seen in Figure 1;second, the variation of SFR over time in the simulated globalSchmidt laws in Figure 8. Nevertheless, as we have demonstratedin the previous sections, our models are valid within one gas con-sumption time �SF and are sufficient to investigate the dominantphysics that controls gravitational collapse and star formationwithin that period.

7.3. Initial Conditions of Galaxies

Many nearby galaxies appear to be gas-poor and stable(Qsg > 1). However, their progenitors at high redshift were gas-rich, so the bulk of star formation should have taken place earlyon. In order to test this, we vary the gas fraction (in terms of totaldisk mass) in the models. We also vary the total galaxy mass andthus the rotational velocity. These result in different initial sta-bility curves. Massive or gas-rich galaxies have low values of theQ parameters, so they are unstable, forming stars quickly andefficiently.

There are no observations yet that directly measure theQ-values in starburst galaxies. However, indirectly, observationsby Dalcanton et al. (2004) show that dust lanes, which trace starformation, only form in unstable regions. Moreover, observa-tions of color gradients in disk galaxies by MacArthur et al.(2004) show that massive galaxies form stars earlier and withhigher efficiency. Both of these observations are naturally ex-plained by our models.

The Toomre Q parameter for gas Qg differs from that for acombination of stars and gas Qsg in some of our model galaxies.This leads to slightly different results in Figures 3 and 10, wherewe compareQsg;min andQg;min. However, we believe thatQsg is abetter measure of gravitational instability in the disk, as it takesinto account both the collisionless and the collisional compo-nents, as well as the interaction between them. We note that itis a simplified approach to quantify the instability of the entiredisk with just a number Qsg;min, as Q has a radial distribution,evolves with time, and is an azimuthally averaged quantity,

but nevertheless, we find interesting regularities by making thisapproximation.

8. SUMMARY

We have simulated gravitational instability in galaxies withsufficient resolution to resolve collapse to molecular cloud pres-sures in models of a wide range of disk galaxies with differenttotal mass, gas fraction, and initial gravitational instability. Ourcalculations are based on two approximations: the gas of thegalactic disk has an isothermal equation of state, representing aroughly constant gas velocity dispersion; and sink particles areused to follow gravitationally collapsed gas, which we assume toform both stars and molecular gas. With these approximations,we have derived star formation histories, radial profiles of thesurface density of molecular and atomic gas and SFR, both theglobal and local Schmidt laws for star formation in galaxies, andthe SFE.The star formation histories of our models show the expo-

nential dependence on time given by equation (6) in agreementwith, for example, the interpretation of galactic color gradientsby MacArthur et al. (2004). The radial profiles of atomic andmolecular gas qualitatively agree with those observed in nearbygalaxies, with surface density of molecular gas peaking centrallyat values much above that of the atomic gas (e.g., Wong & Blitz2002). The radial profile of the surface density of SFR correlateslinearly with that of the molecular gas, agreeing with the ob-servations of Gao & Solomon (2004a).Our models quantitatively reproduce the observed global

Schmidt law (Kennicutt 1998a), the correlation between thesurface density of SFR �SFR and the gas surface density �gas,in both the slope and normalization over a wide range of gassurface densities (eq. [9]). We show that �SFR is strongly cor-related with the gravitational instability of galaxies �SFR /½Qsg;min(�SF)��1:54�0:23, where Qsg;min(�SF) is the local instabilityparameter at time t ¼ �SF (see eq. [13]). This correlation natu-rally leads to the Schmidt law.On the other hand, our models do not reproduce the correla-

tion�SFR � �gas� derived from kinematical models (Kennicutt1998a). However, they may agree better with the dependence ofthe normal galaxies on this quantity, as suggested by Boissieret al. (2003). The discrepancy may be caused by the lack ofextreme starburst galaxies such as galaxy mergers in our set ofmodels.The local Schmidt laws of individual galaxies clearly show

evidence of star formation thresholds above a critical surfacedensity. The threshold surface density varies with galaxy andappears to be determined by the gravitational stability of thedisk. The derived threshold parameters for our stable modelscover the range of values seen in observations of normal gal-axies. The local Schmidt laws have significant variations in bothslope and normalization but also cover the observational rangesreported by Wong & Blitz (2002), Boissier et al. (2003), andHeyer et al. (2004). The average normalization and slope of thelocal power laws are very close to those of the global Schmidtlaw.Our models show that the global SFE �g can be quantitatively

predicted by the gravitational instability of the disk. We haveused a fixed local SFE �‘ ¼ 30% to convert the mass of the sinkparticles to stars in our analysis. This is a reasonable assumptionfor the SFE in dense, high-pressure molecular clouds. The globalSFE of a galaxy then can be shown to depend quantitatively ona nonlinear function (eq. [19]) of the minimum Toomre param-eter Qsg;min for stars and gas that can be approximated for


Qsg;min � 1:0 with the linear correlation �g/�‘ / 0:9� Qsg;min.More unstable galaxies have higher SFE. Massive or gas-richgalaxies in our suite of models are unstable, forming stars quicklywith high efficiency. They represent starburst galaxies. Small orgas-poor galaxies are rather stable, forming stars slowly with lowefficiency, corresponding to quiescent, normal galaxies.

We thank V. Springel for making both GADGET and hisgalaxy initial condition generator available, as well as for usefuldiscussions, and A.-K. Jappsen for participating in the imple-mentation of sink particles in GADGET. We are grateful to

F. Adams, G. Bryan, J. Dalcanton, B. Elmegreen, D. Helfand,R. Kennicutt, J. Lee, C.Martin, R.McCray, T. Quinn, E. Quataert,M. Shara, and J. van Gorkom for very useful discussions. Wealso thank the referee, C. Struck, for valuable comments thathelp to improve this manuscript. This work was supported bythe NSF under grants AST 99-85392 and AST 03-07854, byNASA under grant NAG5-13028, and by the Emmy NoetherProgram of the DFG under grant KL1358/1. Computations wereperformed at the Pittsburgh Supercomputer Center supportedby the NSF, on the Parallel Computing Facility of the AMNH,and on an Ultrasparc III cluster generously donated by SunMicrosystems.

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LI, MAC LOW, & KLESSEN896

STAR FORMATION IN ISOLATED DISK GALAXIES. II. …more, inefﬁcient star formation can be found well outside the threshold radius (Ferguson et al. 1998). What is the origin of the

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